Stability of Cylindrical Transonic Shocks for Two-Dimensional Steady Compressible Euler Flows

Abstract. For given supersonic flow passing a divergent nozzle, if the pressure at the exit of the nozzle (back pressure) is sufficiently large, then a transonic shock may appear in the nozzle. The pressure increases across the shock front, while the supersonic flow jumps down to subsonic. This paper is devoted to analyze such phenomena by establishing the stability of a class of cylindrical symmetric transonic shocks for two–dimensional complete compressible steady Euler system. This result also partly explains the effectiveness of the popular quasi–one–dimensional model of nozzle flows used in aerodynamics. Mathematically, this is to solve a nonlinear free boundary problem for an elliptic–hyperbolic composite system, with the circular transonic shock front as the free boundary. We accomplish this by finding the (locally) unique fixed point of an appropriately defined boundary profile updating mapping. To define this mapping, we encounter a series of nonclassical boundary value problems on an annulus, which involve a new type of nonlocal elliptic problem, and integral–like solvability conditions to determine the position of the free boundary. This reflects an interesting new feature of boundary value problems of elliptic–hyperbolic composite systems. 1